LAGUNA BEACH, Calif.–As our digital trails explode, privacy is being eroded. At the WSJD Live global technology conference Wednesday, two heavyweights in predictive analytics debated whether it’s possible to use data-mining technologies and protect privacy at the same time.

Alex Karp, chief executive of Palantir Technologies Inc., a highly valued startup, conceded that privacy was at risk. “I’m not signing up,” to give outsiders and citizens access to my data, he said. “We live in a society where only people who were...